3.709 \(\int \frac{\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=121 \[ -\frac{\cos ^7(c+d x)}{7 a d}+\frac{\sin (c+d x) \cos ^7(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{48 a d}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{192 a d}-\frac{5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac{5 x}{128 a} \]

[Out]

(-5*x)/(128*a) - Cos[c + d*x]^7/(7*a*d) - (5*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) - (5*Cos[c + d*x]^3*Sin[c +
d*x])/(192*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(48*a*d) + (Cos[c + d*x]^7*Sin[c + d*x])/(8*a*d)

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Rubi [A]  time = 0.145237, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2839, 2565, 30, 2568, 2635, 8} \[ -\frac{\cos ^7(c+d x)}{7 a d}+\frac{\sin (c+d x) \cos ^7(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{48 a d}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{192 a d}-\frac{5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac{5 x}{128 a} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

(-5*x)/(128*a) - Cos[c + d*x]^7/(7*a*d) - (5*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) - (5*Cos[c + d*x]^3*Sin[c +
d*x])/(192*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(48*a*d) + (Cos[c + d*x]^7*Sin[c + d*x])/(8*a*d)

Rule 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\cos ^8(c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int \cos ^6(c+d x) \sin (c+d x) \, dx}{a}-\frac{\int \cos ^6(c+d x) \sin ^2(c+d x) \, dx}{a}\\ &=\frac{\cos ^7(c+d x) \sin (c+d x)}{8 a d}-\frac{\int \cos ^6(c+d x) \, dx}{8 a}-\frac{\operatorname{Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac{\cos ^7(c+d x)}{7 a d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac{\cos ^7(c+d x) \sin (c+d x)}{8 a d}-\frac{5 \int \cos ^4(c+d x) \, dx}{48 a}\\ &=-\frac{\cos ^7(c+d x)}{7 a d}-\frac{5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac{\cos ^7(c+d x) \sin (c+d x)}{8 a d}-\frac{5 \int \cos ^2(c+d x) \, dx}{64 a}\\ &=-\frac{\cos ^7(c+d x)}{7 a d}-\frac{5 \cos (c+d x) \sin (c+d x)}{128 a d}-\frac{5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac{\cos ^7(c+d x) \sin (c+d x)}{8 a d}-\frac{5 \int 1 \, dx}{128 a}\\ &=-\frac{5 x}{128 a}-\frac{\cos ^7(c+d x)}{7 a d}-\frac{5 \cos (c+d x) \sin (c+d x)}{128 a d}-\frac{5 \cos ^3(c+d x) \sin (c+d x)}{192 a d}-\frac{\cos ^5(c+d x) \sin (c+d x)}{48 a d}+\frac{\cos ^7(c+d x) \sin (c+d x)}{8 a d}\\ \end{align*}

Mathematica [B]  time = 11.3751, size = 481, normalized size = 3.98 \[ -\frac{1680 d x \sin \left (\frac{c}{2}\right )-1680 \sin \left (\frac{c}{2}+d x\right )+1680 \sin \left (\frac{3 c}{2}+d x\right )+336 \sin \left (\frac{3 c}{2}+2 d x\right )+336 \sin \left (\frac{5 c}{2}+2 d x\right )-1008 \sin \left (\frac{5 c}{2}+3 d x\right )+1008 \sin \left (\frac{7 c}{2}+3 d x\right )-168 \sin \left (\frac{7 c}{2}+4 d x\right )-168 \sin \left (\frac{9 c}{2}+4 d x\right )-336 \sin \left (\frac{9 c}{2}+5 d x\right )+336 \sin \left (\frac{11 c}{2}+5 d x\right )-112 \sin \left (\frac{11 c}{2}+6 d x\right )-112 \sin \left (\frac{13 c}{2}+6 d x\right )-48 \sin \left (\frac{13 c}{2}+7 d x\right )+48 \sin \left (\frac{15 c}{2}+7 d x\right )-21 \sin \left (\frac{15 c}{2}+8 d x\right )-21 \sin \left (\frac{17 c}{2}+8 d x\right )-336 \cos \left (\frac{c}{2}\right ) (7 c-5 d x)+1680 \cos \left (\frac{c}{2}+d x\right )+1680 \cos \left (\frac{3 c}{2}+d x\right )+336 \cos \left (\frac{3 c}{2}+2 d x\right )-336 \cos \left (\frac{5 c}{2}+2 d x\right )+1008 \cos \left (\frac{5 c}{2}+3 d x\right )+1008 \cos \left (\frac{7 c}{2}+3 d x\right )-168 \cos \left (\frac{7 c}{2}+4 d x\right )+168 \cos \left (\frac{9 c}{2}+4 d x\right )+336 \cos \left (\frac{9 c}{2}+5 d x\right )+336 \cos \left (\frac{11 c}{2}+5 d x\right )-112 \cos \left (\frac{11 c}{2}+6 d x\right )+112 \cos \left (\frac{13 c}{2}+6 d x\right )+48 \cos \left (\frac{13 c}{2}+7 d x\right )+48 \cos \left (\frac{15 c}{2}+7 d x\right )-21 \cos \left (\frac{15 c}{2}+8 d x\right )+21 \cos \left (\frac{17 c}{2}+8 d x\right )-2352 c \sin \left (\frac{c}{2}\right )+4704 \sin \left (\frac{c}{2}\right )}{43008 a d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^8*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

-(-336*(7*c - 5*d*x)*Cos[c/2] + 1680*Cos[c/2 + d*x] + 1680*Cos[(3*c)/2 + d*x] + 336*Cos[(3*c)/2 + 2*d*x] - 336
*Cos[(5*c)/2 + 2*d*x] + 1008*Cos[(5*c)/2 + 3*d*x] + 1008*Cos[(7*c)/2 + 3*d*x] - 168*Cos[(7*c)/2 + 4*d*x] + 168
*Cos[(9*c)/2 + 4*d*x] + 336*Cos[(9*c)/2 + 5*d*x] + 336*Cos[(11*c)/2 + 5*d*x] - 112*Cos[(11*c)/2 + 6*d*x] + 112
*Cos[(13*c)/2 + 6*d*x] + 48*Cos[(13*c)/2 + 7*d*x] + 48*Cos[(15*c)/2 + 7*d*x] - 21*Cos[(15*c)/2 + 8*d*x] + 21*C
os[(17*c)/2 + 8*d*x] + 4704*Sin[c/2] - 2352*c*Sin[c/2] + 1680*d*x*Sin[c/2] - 1680*Sin[c/2 + d*x] + 1680*Sin[(3
*c)/2 + d*x] + 336*Sin[(3*c)/2 + 2*d*x] + 336*Sin[(5*c)/2 + 2*d*x] - 1008*Sin[(5*c)/2 + 3*d*x] + 1008*Sin[(7*c
)/2 + 3*d*x] - 168*Sin[(7*c)/2 + 4*d*x] - 168*Sin[(9*c)/2 + 4*d*x] - 336*Sin[(9*c)/2 + 5*d*x] + 336*Sin[(11*c)
/2 + 5*d*x] - 112*Sin[(11*c)/2 + 6*d*x] - 112*Sin[(13*c)/2 + 6*d*x] - 48*Sin[(13*c)/2 + 7*d*x] + 48*Sin[(15*c)
/2 + 7*d*x] - 21*Sin[(15*c)/2 + 8*d*x] - 21*Sin[(17*c)/2 + 8*d*x])/(43008*a*d*(Cos[c/2] + Sin[c/2]))

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Maple [B]  time = 0.082, size = 551, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x)

[Out]

-2/7/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8+5/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)-2/7/d/a/(1+tan(1/2*
d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^2-397/192/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^3-6/d/a/(1+tan(
1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^4+895/192/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^5-6/d/a/(1+
tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^6-1765/192/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^7-10/d
/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^8+1765/192/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^
9-10/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^10-895/192/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1
/2*c)^11-2/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^12+397/192/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2
*d*x+1/2*c)^13-2/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(1/2*d*x+1/2*c)^14-5/64/d/a/(1+tan(1/2*d*x+1/2*c)^2)^8*tan(
1/2*d*x+1/2*c)^15-5/64/a/d*arctan(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.56373, size = 676, normalized size = 5.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/1344*((105*sin(d*x + c)/(cos(d*x + c) + 1) - 384*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2779*sin(d*x + c)^3/(
cos(d*x + c) + 1)^3 - 8064*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 6265*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 80
64*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 12355*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 13440*sin(d*x + c)^8/(cos
(d*x + c) + 1)^8 + 12355*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 13440*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 6
265*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 2688*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 2779*sin(d*x + c)^13/
(cos(d*x + c) + 1)^13 - 2688*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 - 105*sin(d*x + c)^15/(cos(d*x + c) + 1)^15
- 384)/(a + 8*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 28*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 56*a*sin(d*x
+ c)^6/(cos(d*x + c) + 1)^6 + 70*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 56*a*sin(d*x + c)^10/(cos(d*x + c) +
1)^10 + 28*a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 + 8*a*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 + a*sin(d*x +
c)^16/(cos(d*x + c) + 1)^16) - 105*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

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Fricas [A]  time = 1.13154, size = 189, normalized size = 1.56 \begin{align*} -\frac{384 \, \cos \left (d x + c\right )^{7} + 105 \, d x - 7 \,{\left (48 \, \cos \left (d x + c\right )^{7} - 8 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/2688*(384*cos(d*x + c)^7 + 105*d*x - 7*(48*cos(d*x + c)^7 - 8*cos(d*x + c)^5 - 10*cos(d*x + c)^3 - 15*cos(d
*x + c))*sin(d*x + c))/(a*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**8*sin(d*x+c)/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 1.18302, size = 312, normalized size = 2.58 \begin{align*} -\frac{\frac{105 \,{\left (d x + c\right )}}{a} + \frac{2 \,{\left (105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 2688 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} - 2779 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 2688 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 6265 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 13440 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 12355 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 13440 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 12355 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 8064 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 6265 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 8064 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 2779 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 384 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 384\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{8} a}}{2688 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/2688*(105*(d*x + c)/a + 2*(105*tan(1/2*d*x + 1/2*c)^15 + 2688*tan(1/2*d*x + 1/2*c)^14 - 2779*tan(1/2*d*x +
1/2*c)^13 + 2688*tan(1/2*d*x + 1/2*c)^12 + 6265*tan(1/2*d*x + 1/2*c)^11 + 13440*tan(1/2*d*x + 1/2*c)^10 - 1235
5*tan(1/2*d*x + 1/2*c)^9 + 13440*tan(1/2*d*x + 1/2*c)^8 + 12355*tan(1/2*d*x + 1/2*c)^7 + 8064*tan(1/2*d*x + 1/
2*c)^6 - 6265*tan(1/2*d*x + 1/2*c)^5 + 8064*tan(1/2*d*x + 1/2*c)^4 + 2779*tan(1/2*d*x + 1/2*c)^3 + 384*tan(1/2
*d*x + 1/2*c)^2 - 105*tan(1/2*d*x + 1/2*c) + 384)/((tan(1/2*d*x + 1/2*c)^2 + 1)^8*a))/d